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17
votes
6answers
2k views

Parametrization of the boundary of the Mandelbrot set

Does anyone know how to parametrize the boundary of the Mandelbrot set? I am not a fractal-geometer or a dynamical systems person. I just have some idle curiosity about this question. The ...
8
votes
2answers
507 views

Error of Discrete Fourier Transform on Finite Domain (vs. Continuous FT) in terms of Sobolev order

My question is about quantifying the error that occurs by approximating the continuous Fourier transform on a finite domain by using a discretised version with resolution $N$ for a function of a given ...
1
vote
0answers
62 views

Ideal structure of group $C^*$-agebras [on hold]

What is the ideal structure of group $C^*$-algebras? Do there exist any books or articles in the field ? If G to be the group of integers $Z$ , then $C^*$($Z$)= C($T$) so because ideal structure of $ ...
1
vote
1answer
59 views

harmonic balance method for a 2-mass 3-spring system

I am trying to solve a nonlinear 3spring-2mass system under harmonic loading by using Fourier series expansion of states of the differential equation. The system is just basically two masses, two ...
2
votes
1answer
86 views

The inverse of Laplacian operator for different orders

I post this question in MSE couple of days before and get no response. So I repost it here for better luck. Thank you! Let $u,v\in C_c^\infty(\Omega)$ and $\Omega\subset \mathbb R^N$ is open ...
2
votes
0answers
65 views

Multiplier operators on anisotropic weighted $L^2$ spaces

Suppose $\mathcal{M}$ is a multiplier operator on $L^2(\mathbb{R})$, in the sense that, for any $u(x)\in L^2(\mathbb{R})$, $\widehat{\mathcal{M}u}(k)=m(k)\hat{u}(k),$ where the scalar complex function ...
2
votes
1answer
204 views

Decoupling in mixed norm spaces

Bourgain and Demeter's proof of the $L^2$ decoupling conjecture decouples $\|f\|_{L^p}$ into an $L^2$ sum of $\|f_\theta\|_{L^p}$, where $\hat f$ is supported on a curved hypersurface $S$, where ...
0
votes
0answers
35 views

Necessary condition for decouplings for surfaces in $\mathbb{R}^4$

I'm currently studying the paper Decouplings for surfaces in $\mathbb{R}^4$ written by Bourgain and Demeter. This paper is available in here. As an example of nondegenerate $2$-dimensional surfaces ...
0
votes
0answers
33 views

Kernel for projection operators onto the spaces of piecewise linear loops

For every $m\in\mathbb{N}_+$, $k=0,\dots,m-1$, denote $I_{m,k}:=(\frac{k}{m},\frac{k+1}{m})$. Denote $W = H^1(S^1,\mathbb R^{2n})$ = The Hilbert space of $C^1$ maps maps $v(t)$ from the circle to ...
0
votes
0answers
57 views

What's the relationship between the roots of a function and that of a filtered Fourier series representation?

Suppose $M$ is a piecewise constant function on an interval $T$ taking values $+1$ and $-1$, and that $M$ exhibits all the properties sufficient to ensure the existence of some converging Fourier ...
0
votes
0answers
70 views

Resolvent of the operator

Consider the Laplace operator defined on the biggest possible subset of$L^{2}(R^{2})$: $T= - \partial^{2}_{x} -\partial^{2}_{y}+x^{2}+y^{2}+ 2.i(x \frac{\partial}{\partial ...
26
votes
2answers
3k views

Is this Riemann zeta function product equal to the Fourier transform of the von Mangoldt function?

Mathematica knows that: $$\log(n) = \lim\limits_{s \rightarrow 1} \zeta(s)\left(1 - \frac{1}{n^{(s - 1)}}\right)\;\;\;\;\;\;\;\;\;\;\;\; (1)$$ The von Mangoldt function should then be: ...
9
votes
2answers
319 views

Completeness of nonharmonic Fourier Series

I have the following question: The Exponential System $(\exp(2\pi i n \cdot ))_{n\in \mathbb{Z}}$ constitutes an orthonormal basis of $L^2([-1/2,1/2])$. Thus, certainly the oversampled system ...
8
votes
2answers
502 views

Can exponential sums be small on a whole interval?

This is almost certainly routine to an analyst, so forgive me in advance. Let $\alpha_i\in \mathbb{R}$. Consider the functional $$\varphi: L^1[0.9A,A]\to \mathbb{C}$$ via $$f\mapsto \sum_i ...
2
votes
1answer
39 views

What are the known conditions for the log of the Fourier transform of a 2D real discrete signal to have no branch cuts?

Suppose we sample a real 2D signal, f(x,y), at N by N evenly spaced points in x and y. Then we compute the Fourier transform of the sampled signal, F(u,v), and then take the log of F. There will be a ...
2
votes
1answer
251 views

To understand integral :$\lambda (x) = \int_{0}^{\infty} \frac{\sin^{2} \alpha x}{\alpha^{2}} d\mu(\alpha), (\mu(0)=0)$

I wants to understand the integrals of the form $$\lambda (x) = \int_{0}^{\infty} \frac{\sin^{2} \alpha x}{\alpha^{2}} d\mu(\alpha), (\mu(0)=0)$$ where $\mu(\alpha)$ is a non decreasing function ...
0
votes
0answers
72 views

The growth rate of almost periods for almost periodic function

A subset $A \subset \mathbb{R}^2$ is relative dense if there exists $L>0$ such that for every $p\in \mathbb{R}^2$ there exists $p' \in A$ such that $|p-p'|<L.$ A continuous function $f : ...
8
votes
2answers
449 views

Interpret Fourier transform as limit of Fourier series

Let $V=\mathbb{R}^n$, $\Lambda_r=2\pi r \mathbb{Z}^n \subset V (r>0)$ a lattice; $V^*\cong\mathbb{R}^n$ the dual vector space of $V$, and $\Lambda_r^*=\frac{1}{2\pi r} \mathbb{Z}^n ...
2
votes
1answer
112 views

Solving a simple Schrödinger equation with Fast Fourier Transforms

While trying to solve a stochastic Gross-Pitaevskii equation I have found a problem that can be tracked down to something buggy occurring in the simplest Schrodinger equation possible: $$\partial_t ...
8
votes
2answers
283 views

Is the Fourier transform of $e^{-|x|^n}$ positive?

Let $$\Phi(x) = \int_{\mathbf{R}^n} e^{-|y|^n +i (x,y)} dy.$$ Is $\Phi$ positive everywhere in $\mathbf{R}^n$? Could someone helps me answer this question or gives a reference for it? Thanks.
5
votes
1answer
248 views

Dominated convergence to characteristic function

Let $\phi_m(x):=\chi_{[0,1]} * \chi_{[0,1]} *...* \chi_{[0,1]}$ be the m -times convolution (so $m+1$ characteristic functions are involved). Then the Fourier transform of this function is given by ...
0
votes
0answers
43 views

Matrix representation of linearized PDE

My motivation for this question is to investigate the linear stability of steady solutions of a nonlinear PDE by computing eigenvalues of the linearized equations. Specifically, I have a function ...
36
votes
4answers
3k views

What are fixed points of the Fourier Transform

The obvious ones are 0 and $e^{-x^2}$ (with annoying factors), and someone I know suggested hyperbolic secant. What other fixed points (or even eigenfunctions) of the Fourier transform are there?
1
vote
1answer
48 views

Seeking a specific proof of endpoint boundedness of Riesz potential

The Riesz potential is defined by $$I_\alpha (f)(x) := c_{d, \alpha} \int_{\mathbb R^d} \frac{f(y)}{|x-y|^{d-\alpha}} \, dy.$$ Once $f\in L^{d/\alpha}(\mathbb{R}^n)$, then $I_\alpha f(x)\in BMO$. ...
2
votes
0answers
85 views

system of complex number equations

Let $a_1,a_2,a_3,a_4\in \mathbb{C}$ be distinct such that $$a_1^3+a_2^3+a_3^3+a_4^3=0$$ $$(1+|a_1|^2)a_1^2+(1+|a_2|^2)a_2^2+(1+|a_3|^2)a_3^2+(1+|a_4|^2)a_4^2=0$$ ...
7
votes
1answer
3k views

Eliminating Gibbs phenomenon, and approximating with jumping functions in Fourier Analysis : An attempt and a question in this regard

This problem seems like a nightmare to me. I tried to expand $K_{\omega}^f(t)$, but I am clueless of getting some kind of a closed form or some kernel like structure. If I try to take ...
19
votes
2answers
642 views

Uncertainty principle for Mellin transform

Let $f:\mathbb{R}^+\to \mathbb{C}$. Let $Mf$ be its Mellin transform: $Mf(s) = \int_0^\infty f(x) x^{s-1} dx$. (a) Some time ago, I convinced myself that $f(t)$, $Mf(\sigma+it)$ and $Mf(\sigma-it)$ ...
0
votes
0answers
33 views

Supremum distribution of band-limited functions with random spectrum

Consider the properties of band-limited functions $f_N:[-\pi,\pi]\to\mathbb{R}$ defined through their Fourier series $f_N(x)=\sum_{n=-N}^N c_n e^{inx}$ where $c_n=a_n+i b_n$ and both ...
5
votes
0answers
105 views

Nonclassical polynomials, circles, and groups

Tao and Ziegler have introduced a generalization of polynomials over a prime field called nonclassical polynomials, useful for studying the Gowers norm. A nonclassical polynomial of degree $d$ is a ...
8
votes
2answers
436 views

Fourier transform of the critical line of zeta?

This was asked on MSE and got a lot of upvotes but no answers, so I'm posting it here. Is there a known expression for the (distributional) Fourier transform of the Riemann zeta function, taken along ...
8
votes
0answers
174 views

Precise relationship between “finite” Fourier analysis and Galois theory in solving the cubic?

Suppose we want to solve$$x^3 - ax^2 + bx - c = 0.$$We know a priori that this can be factored as $(x - r_0)(x - r_1)(x - r_2)$; by Vieta's formulas, we know$$a = r_0 + r_1 + r_2,\quad b = r_0r_1 + ...
9
votes
1answer
395 views

Absolute continuity reflected in Fourier coefficients?

Imagine $\mu$ and $\nu$ are two Borel probabilty measures in the interval $[0,1]$. We say that $\mu$ is absolutely continuous with respect to $\nu$, if for every measurable set $A$ such that ...
-1
votes
1answer
134 views

separable BV space for PDE's, Whats stopping us? [closed]

Consider the metric space BV(0,1) with the following metric $$ d(u,v) = \|u-v\|_{L^1} + |TV(u)-TV(v)| $$. It is separable, compact, uniformly bounded and complete. So What is the really obvious thing ...
1
vote
2answers
232 views

Fourier transform localisation (still unanswered, but apparently off-topic?) [closed]

In the context of Pólya's theorem I was reading these notes here on p. 19. In the last paragraph the authors claim (it is the sentence starting like "standard Fourier theory shows...") that the ...
2
votes
0answers
342 views

A problem about Joint sine and cosine fourier transform

There is a problem on Page 60 of Book of B.M.Budak, A. A. Samarskii and A. N. Tikhonov, whose title is A Collection of Problems in Mathematical Physics (New York, Dover, 1964). The problem (the ...
0
votes
0answers
67 views

Integrating a series expansion of $\mbox{frac}(x)\lfloor x\rfloor$ coming from Fourier series of sawtooth function

Let me preface this question by saying that I am not exactly sure it counts as research level. It is crossposted on mathstackexchange: ...
1
vote
1answer
106 views

A metric on the set of BV functions, is it mentioned/studied in literature?

I'd like to propose the following metric which operates on the set $M$ of all square integrable functions that are also of bounded variation, of the form $f : (0,1) \to \mathbb{R}$. Given any $x,y \in ...
3
votes
0answers
134 views

Proof without distributions

I was wondering whether there is a way to show this identity $$\pi \int_{\mathbb{R}^3} \frac{f(x)}{|x|} dx = \int_{\mathbb{R}^3} \frac{\widehat{f(x)}}{|x|^2} dx $$ without using distributions for $f ...
2
votes
1answer
91 views

Lower bounds from Fourier dimension?

According to Mattila, Geometry of sets and measures in Euclidean spaces, p. 168, the Fourier dimension $\text{dim}_F(A)$ of $A\subseteq \mathbb R^n$ is the unique number in $[0,n]$ such that for any ...
3
votes
2answers
297 views

Bandwidth approximation for a nonlinear problem

Can anyone please help me with this problem. I must let you know from the beginning that it's not an easy one. "Two functions are given: $u, y \in L^{2}(-\infty,\infty), y(t)=\frac{u(t)}{u(t)+b}$ , ...
0
votes
0answers
179 views

Express measurable entropy in terms of Fourier coefficients of the measure

Let $S^1$ be the unit circle and $T:S^1\to S^1$ be a continuous map. Suppose $\mu$ is a $T$-invariant Borel probability measure on $S^1$, that is, $\mu(T^{-1}A)=\mu(A)$ for every Borel subset $A$ of ...
2
votes
1answer
852 views

Is the integral always nonzero?

Let $$I_{n,p,a}:=\int_0^{\infty-} \frac{g_{n,a}(t)}{t^{p+1}} \, dt,$$ where $$(*)\qquad\qquad\qquad n\in\mathbb N,\quad -\infty<a<\infty,\quad p_{n,a} < p < ...
2
votes
0answers
66 views

Combining oscillatory integrals of the first and second kind

Consider an oscillatory integral of the first kind $$ I_\lambda(x)=\intop_{\mathbb{R}^{n}}e^{i\lambda\Phi(x,y)}a(x,y)\,d y,\quad \lambda\geq 0,\; a\in C_c^\infty(\mathbb{R}^{k+n}),\; \Phi\in ...
9
votes
3answers
2k views

English translation of Riemann's Habilitation Thesis

Does anyone know where to find an English translation of Riemann's Habilitation Thesis concerning trigonometric series? The German title of the work is "Über die Darstellbarkeit einer Function durch ...
7
votes
1answer
264 views

Maximal $L_1$ norm of Fourier Transform of a Subset

Let $A_n$ be the following $n\times n$ matrix: $(A_n)_{i,j}= \frac{1}{\sqrt{n}}\omega_n^{i\cdot j}$ for all $0 \le i,j <n$, where $\omega_n=e^{\frac{2\pi i}{n}}$. I want to understand how $A_n$ ...
2
votes
0answers
88 views

Error term for a Fourier integral

There is a well-known theorem that states that for $f$ continuous and $f,\hat f$ integrable, $$f(0)=\frac{1}{\pi}\lim_{T\to\infty}\int_{-\infty}^\infty f(x)\frac{\sin(Tx)}{x}dx.$$ So it should be that ...
4
votes
0answers
106 views

Symmetry of Sundry Planar Convex Sets of Constant Width & Minimal Area

In a much broader paper in “Optimization Methods & Software 27,6 (2012) pp1073-1099” Bayen & Henrion consider planar compact, convex sets with support functions which are finite Fourier ...
3
votes
0answers
91 views

Does the following inequality hold under Zygmund condition?

Let $f:R\rightarrow R$ be a continuous function which satisfies the Zygmund condition $$ |f(x+\delta)+f(x-\delta)-2f(x)|\leq \text{const} \frac{|\delta|}{(\log\frac{1}{|\delta|})^{\alpha}}, \,\, ...
0
votes
0answers
83 views

Proof sought that a certain function is approximatively 3-dimensional

Let $D_n(x)$ be a diagonal matrix of size $N\times N$ where the $k$th element is $\exp(2\pi\jmath x(n+(k-1)/N)$. Let $P_n$ be a random diagonal $N\times N$ matrix where each diagonal element is a ...
0
votes
0answers
70 views

Is there a space in which the $\vec a$ in $\sin(a_1\cdot x)+\sin(a_2\cdot x)$ is linear?

Suppose one has equations of the form $\sin(a_1\cdot x_i)+\sin(a_2\cdot x_i)=y_i$ for $i = 1, \dots, n$ (there are also amplitudes and phase shifts, but let's ignore these for now). I want to solve ...